definition of locally closed topological subspace
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological subspace.
- The reader knows a definition of closed subset of topological space.
- The reader knows a definition of neighborhood of point on topological space.
Target Context
- The reader will have a definition of locally closed topological subspace.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( T'\): \(\in \{\text{ the topological spaces }\}\)
\(*T\): \(\subseteq T'\), with the subspace topology of \(T'\)
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Conditions:
\(\forall t \in T (\exists U'_t \in \{\text{ the open neighborhoods of } t \text{ on } T' \text{ with the subspace topology of } T'\} (U'_t \cap T \in \{\text{ the closed subsets of } U'_t\}))\)
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2: Note
When \(T \subseteq T'\) is closed, \(T\) is locally closed, because for whatever open neighborhood of \(t\), \(U'_t \subseteq T'\), for example, \(U'_t = T'\), \(U'_t \cap T\) is closed on \(U'_t\), by the proposition that any subset on any topological subspace is closed if and only if there is a closed set on the base space whose intersection with the subspace is the subset.
When \(T \subseteq T'\) is open, \(T\) is locally closed, because there is an open neighborhood of \(t\), \(U'_t \subseteq T'\), such that \(U'_t \subseteq T\), by the local criterion for openness, and \(U'_t \cap T = U'_t \subseteq U'_t\) is closed.
Any interval, \(T = I \subseteq \mathbb{R} = T'\), where \(I = (r_1, r_2), (r_1, r_2], [r_1, r_2), \text{ or } [r_1, r_2]\) where \(r_1\) may be \(- \infty\) when \(I\) is lower open and \(r_2\) may be \(\infty\) when \(I\) is upper open, is locally closed, because while any open or closed case is already known to be locally closed, for \((r_1, r_2]\), when \(t \neq r_2\), there is a \(U'_t \subseteq (r_1, r_2]\), and \(U'_t \cap T = U'_t \subseteq U'_t\) is closed, and when \(t = r_2\), there is a \(U'_t = B_{r_2, \epsilon}\) such that \(r_1 \lt r_2 - \epsilon\), and \(U'_t \cap T = (r_2 - \epsilon, r_2]\), which is closed on \(B_{r_2, \epsilon}\), because \((r_2 - \epsilon, r_2] = B_{r_2, \epsilon} \cap [r_2 - \epsilon, r_2]\); for \([r_1, r_2)\), likewise.
Then, what are not locally closed?
For example, \(T = \mathbb{Q} \subseteq \mathbb{R} = T'\) is not locally closed, because for any \(q \in \mathbb{Q}\), for any \(U'_q \subseteq \mathbb{R}\), \(U'_q \cap \mathbb{Q} \subseteq U'_q\) is not closed, because \(U'_q \setminus (U'_q \cap \mathbb{Q}) \subseteq U'_q\) is not open, because for each \(r \in U'_q \setminus (U'_q \cap \mathbb{Q})\), any \(B'_{r, \epsilon} \subseteq \mathbb{R}\) such that \(B'_{r, \epsilon} \subseteq U'_q\) contains a rational number, so, \(B'_{r, \epsilon} \subseteq U'_q \setminus (U'_q \cap \mathbb{Q})\) does not hold.
